Consider a market with two agents: a seller and a buyer. The seller holds an item for sale and the buyer considers buying this item. The seller values the item as and the buyer values it as . The utilities of both agents are quasi-linear in money. If then there is a potential gain-from-trade (GFT) — if the seller sells the item to the buyer then the social welfare will increase by . We look for a mechanism that asks the agents to report their values and implements this optimal GFT. The mechanism should satisfy four properties:
Efficiency — the increase in social welfare should be .
Budget balance (BB) — the mechanism designer should not lose.
Individual rationality (IR) — the utility of each agent should not decrease.
Dominant-strategy incentive-compatibility (DSIC) — it should be a weakly-dominant strategy for each agent to report his true value.
It is easy to construct mechanisms that satisfy each three of these four properties. However, a classic paper of Myerson and Satterthwaite (1983) proves that it is impossible to satisfy all four properties simultaneously. Intuitively, the reason is that it is impossible to truthfully determine prices for trading: if the mechanism pays the seller , the seller is incentivized to bid e.g. to force the price up; similarly, if the mechanism charges the buyer , the buyer is incentivized to bid to force the price down. DSIC can be guaranteed by setting and (this is the two-sided variant of the Vickrey auction (Vickrey, 1961)) but this mechanism is not BB since it leaves the mechanism designer with a deficit of .
Now consider a market in which a price is fixed exogenously, e.g. by the government. Fixing a price automatically guarantees budget-balance, since all monetary transfers are between buyers and sellers. We are interested in incentive-compatibility and efficiency subject to individual-rationality, i.e, we want the buyer to buy the item if-and-only-if . In this case the Myerson–Satterthwaite impossibility disappears. It is trivial to attain all these properties using the following simple mechanism: ask the agents to report their values and perform the trade iff .
But this trivial mechanism works only when there is a single unit of a single good. We are interested in the following question:
In a market with a single buyer, a single seller and two or more items with exogenously-fixed prices, is it possible to attain incentive-compatibility and efficiency subject to individual-rationality?
Our main result is that this is impossible. We quantify the impossibility using the notion of a competitive ratio of a mechanism — the worst-case ratio of its GFT to the optimal GFT. We present three upper bounds on the competitive ratio:
If the seller holds different kinds of goods and the buyer wants one good, then the competitive ratio of any DSIC mechanism is at most .
If the seller holds units of the same good, then the competitive ratio of any DSIC mechanism is at most .
If the seller holds units of the same good, and it is public knowledge that both the seller and the buyer have submodular valuations (decreasing marginal returns), then the competitive ratio of any DSIC mechanism is at most , where is the -th harmonic number ().
All these upper bounds are tight: they are attained by trivial mechanisms that ignore the agents’ bids and determine the trade randomly.
The results are proved as special cases of a general model about bilateral cooperation. This model is presented in Section 2. The results on cooperation with general valuations are proved in Section 3. The result on cooperation with submodular valuations is proved in Section 4. The corollaries to bilateral trading are proved in Section 5.
There are two agents that we call “buyer” and “seller”. They consider options for cooperation, numbered . If option is selected, then the buyer’s utility is and the seller’s utility is , which can be positive zero or negative. These utilities are their private information. Additionally, there is an option 0 which means no cooperation; we assume that .
An outcome is called feasible if both and . Denote the set of feasible outcomes by Feasible:
Note that is never empty since it contains 0.
The optimal utility of a feasible outcome is denoted by OPT:
Bilateral trade is a special case of this general model.
A seller holds units of a single good and values each bundle of units as ; a potential buyer values each bundle of units as . The price-per-unit is and both agents have utility functions quasi-linear in money. Then, and and OPT is the maximum potential GFT (since we normalize the seller’s utility from no cooperation to zero).
A seller holds a set containing different goods, a single unit of each good. The seller has an arbitrary valuation function over bundles of goods: is the seller’s value for bundle . The buyer has a unit-demand valuation function: he values each item as and is interested in buying at most one item. The price of item is fixed at . Then, and and OPT is again the maximum potential GFT.
Our goal is to design a mechanism that approximates OPT.
A mechanism is a function that receives as input the reported valuations of the agents, . These reports may be different than the true valuations . The output of the mechanism,
, is a vector ofprobabilities — a probability for each option including option 0. The probability of option is denoted by , and .
The expected gain of the buyer is:
and the expected gain of the seller is:
and the expected gain of the mechanism is:
A mechanism is called Dominant-Strategy Incentive-Compatible (DSIC) — if for each agent, revealing the true values is a weakly-dominant strategy:
If the mechanism is DSIC, we assume that and , so the mechanism gain is . The competitive ratio of a DSIC mechanism is:
3 General utilities
Consider the following simple mechanism.
The Uniform-Random (UR) mechanism chooses an option with uniform probability, for all , regardless of and . It then lets the agents decide whether they want option or nothing. The agents will cooperate iff . Therefore, the expected gain is
This lower bound is tight, since it is possible that that all positive gain comes from a single option. Therefore, .
Our first theorem shows that, when the agents can have general utility functions, no DSIC mechanism can be better than the uniform-random mechanism.
When there are options, every DSIC mechanism has an expected competitive ratio of at most .
The following lemma will be used several times in the proof.
Let be a DSIC mechanism with competitive ratio . Then, for every :
Consider the alternative report vector , where is a very large number (sufficiently large such that for all ). Note that . If the true values are , then almost all gain comes from the buyer’s values, so:
Since the competitive ratio is , we must have . Dividing both sides by yields:
If the buyer’s true valuations are but he reports , his expected gain is exactly the left-hand side of (*). Therefore, a DSIC mechanism must guarantee that the buyer gets the same utility by reporting truthfully . Hence:
as claimed. The proof for the seller is analogous. ∎
We now apply Lemma 3.3 to utility vectors from a special family. For every , let be the family of vectors with:
For every constant , let be a vector with:
Let be a DSIC mechanism with competitive ratio . Then, for every , every and every :
The proof is by induction on .
For , we have and all other ’s are zero or negative. Apply Lemma 3.3 to the buyer:
We now assume the claim is true for and prove it is true for . Let be any vector in . Consider the alternative buyer’s valuation , with:
So the gain in options is dominated by . We claim that the probability with which the mechanism selects one of these options is negligible. Let . Then, when the true valuations are :
If the competitive ratio of is positive, then must be positive, which implies .
Now, apply Lemma 3.3 to the seller:
Note that . Hence, by the induction assumption:
Summing the above two inequalities gives:
If the buyer’s true valuation is but he deviates and reports , he is guaranteed a utility of at least . Therefore, a DSIC mechanism must guarantee the buyer at least the same utility from reporting truthfully . Hence:
verifying the induction claim for . ∎
4 Submodular utilities
In this section, we assume that the utility vectors of both agents satisfy the condition of decreasing marginal returns, which is equivalent to submodularity:
A utility vector is submodular if, for every :
In the case of multi-unit bilateral trade (Example 2.1), it is clear that if the buyer’s utility function is submodular, then his utility vector is submodular too. It can be checked that the same is true for the seller: if is submodular then is submodular too. We now present a randomized mechanism for submodular agents.
The Decreasing Random (DR) mechanism chooses each option with probability:
where is the -th harmonic number:
It then lets the agents decide whether they want option or nothing. The agents will cooperate iff . Therefore, the expected gain is:
Suppose both the buyer and the seller have submodular utilities. Let . Then the vector is submodular too, and this implies that for every :
Therefore, for every :
This is true in particular for the option in which OPT is attained. Therefore, .
This inequality is tight. For example, we might have, for all , and (both are submodular). Then, the optimal gain is while . Therefore, .
Our next theorem shows that, even when the agents’ utilities are known to be submodular, no DSIC mechanism can be better than the decreasing-random mechanism.
When there are options, every DSIC mechanism that works for arbitrary submodular valuations has an expected competitive ratio of at most .
We will prove the theorem by applying Lemma 3.3 to utility vectors from a special family. For every , let be the family of submodular vectors with:
Let be a vector with: for all (note that is submodular too).
Let be a DSIC mechanism with competitive ratio . Then, for every and for every :
The proof is by induction on .
For , we have and . Applying Lemma 3.3 to the buyer gives .
We now assume the claim is true for and prove it is true for . Let be any vector in . Consider the alternative vector , with:
where is a very small number. By arguments similar to Lemma 3.4, when is sufficiently small, any mechanism with positive competitive ratio must select any option with negligible probability. So, applying Lemma 3.3 to the seller gives:
Since is submodular, and the number we subtract from every grows with , is submodular too. Moreover,
Therefore, and we can apply the induction assumption:
If the buyer’s true valuation is but he deviates and reports , his utility is the left-hand side of (3). A DSIC mechanism must guarantee the buyer at least the same utility from reporting truthfully . Hence:
verifying the induction claim for . ∎
5 Bilateral Trade
In this section we prove two corollaries of Theorem 3.2 for two settings of bilateral trading.
A seller holds units of a single good. The seller values each bundle of units as and a potential buyer values each bundle of units as . The price-per-unit is fixed exogenously. Both agents have utility functions quasi-linear in money. Then, the competitive ratio of any DSIC mechanism is:
at most with general valuations, and:
at most with submodular valuations.
Obviously there are special cases in which Corollary 5.1 does not hold. For example, if it is public knowledge that the utilities of both the buyer and the seller are additive (i.e, and for all ), then the following mechanism is DSIC: ask the agents to report and ; trade units iff .
A seller holds a set containing different goods, a single unit of each good. The seller has an arbitrary valuation function over bundles of goods: is the seller’s value for bundle . The buyer has a unit-demand valuation function: he values each item as and is interested in buying at most one item. Then, the competitive ratio of any DSIC mechanism is at most .
Apply Theorem 3.2 with and . ∎
Corollary 5.3 holds even if it is public knowledge that the seller has an additive valuation function. However, it does not hold if both the seller and the buyer are additive. In this case, it is possible to trade in each item-type separately using a simple mechanism: trade iff .
We presented impossibility theorems on impossibility of truthful cooperation in general. Applying these theorems to Examples 2.1 and 2.2 yields impossibility results on bilateral trade with fixed prices.
Our general results about impossibility of cooperation can potentially be extended in two ways.
1. We assumed that the mechanism is not allowed to charge money — so it must be strongly-budget-balanced. What happens if we allow the mechanism to charge or pay money, but require that it remains weakly-budget-balanced (no deficit)?
2. We required the mechanism to be dominant-strategy IC. What happens if we allow it to be IC only in Bayesian-Nash equilibrium, like the original theorem of Myerson and Satterthwaite (1983)?
The applications to bilateral trade can also be extended. Suppose there are different kinds of goods and the buyer is not unit-demand? If the valuations are arbitrary, then our general impossibility implies that the competitive ratio can be at most . What is the upper bound when the agents’ valuations are sub-additive? sub-modular? gross-substitute?
- Myerson and Satterthwaite (1983) Myerson, R. B. and Satterthwaite, M. A. (1983). Efficient mechanisms for bilateral trading. JET, 29(2):265–281.
- Vickrey (1961) Vickrey, W. (1961). Counterspeculation, Auctions, and Competitive Sealed Tenders. J Finance, 16(1):8–37.